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NS SATURATED AND ${\Delta }_{1}$-DEFINABLE

Part of: Set theory

Published online by Cambridge University Press:  16 February 2021

STEFAN HOFFELNER
STEFAN HOFFELNER*
Affiliation:
WESTFÄLISCHE WILHELMS UNIVERSITÄT, MÜNSTER MÜNSTER, GERMANYE-mail: stefan.hoffelner@gmx.at
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Abstract

We show that under the assumption of the existence of the canonical inner model with one Woodin cardinal $M_1$ , there is a model of $\mathsf {ZFC}$ in which $\mbox {NS}_{\omega _{1}}$ is $\aleph _2$ -saturated and ${\Delta }_{1}$ -definable with $\omega _1$ as a parameter which answers a question of S. D. Friedman and L. Wu. We also show that starting from an arbitrary universe with a Woodin cardinal, there is a model with $\mbox {NS}_{\omega _{1}}$ saturated and ${\Delta }_{1}$ -definable with a ladder system $\vec {C}$ and a full Suslin tree T as parameters. Both results rely on a new coding technique whose presentation is the main goal of this article .

Keywords

nonstationary idealdefinabilitycodinginner modelslarge cardinals

MSC classification

Primary: 03E35: Consistency and independence results
Secondary: 03E45: Inner models, including constructibility, ordinal definability, and core models 03E55: Large cardinals 03E57: Generic absoluteness and forcing axioms
Type
Article
Information
The Journal of Symbolic Logic , Volume 86 , Issue 1 , March 2021 , pp. 25 - 59
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Copyright
© The Association for Symbolic Logic 2021

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